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sophies are violently opposed. It seems unlikely that both can
be as wholly right as each appears to believe he is.
Hilbert returned to Greece for the beginning of his philosophy
of mathematics. Resuming the Pythagorean programme of a
rigidly and fully stated set of postulates from which a mathe-
matical argument must proceed by strict deductive reasoning,
Hilbert made the programme of the postulational development
of mathematics more precise than it had been with the Greeks,
and in 1899 issued the first edition of his classic on the founda-
tions of geometry. One demand which Hilbert made, and which
the Greeks do not seem to have thought of, was that the
proposed postulates for geometry shall be proved to be self-
consistent (free of internal, concealed contradictions). To
produce such a proof for geometry it is shown that any contra-
diction in the geometry developed from the postulates would
imply a contradiction in arithmetic. The problem is thus
shoved back to proving the consistency of arithmetic, and there
it remains to-day.
Thus we are back once more asking the sphinx to tell us what
a number is. Both DedeMnd and Frege fled to the infinite -
Dedekind with his infinite classes defining irrationals, Frege
with his class of all classes similar to a given class defining a
cardinal number - to interpret the numbers that puzzled
Pythagoras. Hilbert, too, would seek the answer in the infinite
which, he believes, is necessary for an understanding of the
finite. He is quite emphatic in his belief that Cantorism will
ultimately be redeemed from the purgatory in which it now
tosses. 'This [Cantor's theory] seems to me the most admirable
fruit of the mathematical mind and indeed one of the highest
achievements of man's intellectual processes.' But he admits
that the paradoxes of Burali-Forti, Russell, and others are not
resolved. However, his faith surmounts all doubts: 'No one
shall expel us from the paradise which Cantor has created for
But at this moment of exaltation Brouwer appears with
something that looks suspiciously like a flaming sword hi Ms
strong right hand. The chase is on: Dedekind, in the role of
Adam, and Cantor disguised as Eve at his side, are already